I was reading in MSE that, up to isomorphism, there are 2 groups of order 45. How do we know that? Is there any way of calculating how many groups of order 10,15 etc. exist up to isomorphism?
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Do you know Sylow theorems in group theory? – DonAntonio Jun 02 '16 at 09:54
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Yes. I know three theorems of sylow and that means only one sylow 3 and sylow 5 subgroup. – low iq Jun 02 '16 at 10:40
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But I am asking a general way.would sylow theorem be enough for all orders? – low iq Jun 02 '16 at 10:40
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Or "special orders " only? – low iq Jun 02 '16 at 10:42
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It's easy to count the abelian groups of this order by the classification of finite abelian groups. From there, it suffices to show that there are no non-abelian groups of this order. – Ben Grossmann Jun 02 '16 at 11:09
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This is tangentially relevant. – Ben Grossmann Jun 02 '16 at 11:17
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A lot depends on the order. No one knows how many groups there are of order $2^{16}$, but it's easy to find the number of groups of order $2^{16}+1$. – Gerry Myerson Jun 02 '16 at 12:44
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@ Gerry myerson And what is that way? – low iq Jun 02 '16 at 14:25
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@low, it doesn't work if you leave a space between the at-sign and the name. But $2^16+1$ is prime, so there's only one group (up to isomorphism) of that order. – Gerry Myerson Jun 03 '16 at 13:14
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@Gerry myerson we don't have any way to know if that's prime. Do we? – low iq Jun 03 '16 at 16:06
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Given any positive integer, there are plenty of ways to know whether or not it is prime. But this particular number, $2^{16}+1$, we know it's prime if we know our math history. It is a "Fermat prime" – look it up! – Gerry Myerson Jun 03 '16 at 22:50
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@Gerry myerson I know this number is prime. I mean to say is there any general way to know if a large number is prime or not – low iq Jun 04 '16 at 01:49
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@low, there is a whole industry devoted to primality testing. People have written books on the topic, so I won't write one here, but suggest you check out the literature on the topic. Well, maybe I can put it this way: whether you can know whether a large number is prime or not depends on how large the number is, on other characteristics of the number, and on the resources at your disposal. – Gerry Myerson Jun 04 '16 at 02:02
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@Gerry myerson how would you know this is prime ----982,451,653 – low iq Jun 05 '16 at 01:46
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@low, I suggested you check out the literature on primality testing. Have you done that? – Gerry Myerson Jun 05 '16 at 12:23
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@Gerry myerson I read about Miller Rabin test in Wikipedia and saw video in you tube.it appears more like algorithms which are not handy often. – low iq Jun 05 '16 at 12:39
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Keep reading. There are plenty of good algorithms. With the right hardware and software, it is routine nowadays to test 1000-digit numbers for primality. – Gerry Myerson Jun 05 '16 at 12:42
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@Gerry myerson deleted – low iq Jun 06 '16 at 00:05
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As you say, by Sylow theorems a group $\;G\;$ of order $\;45\;$ has one unique subgroup $\;P\;$ or order $\;3^2=9\;$ and one unique subgroup $\;Q\;$ of order $\;5\;$ , which means $\;P,\,Q\lhd G\;$ . Also, both $\;P,\,Q\;$ are abelian and so is their product, which generates their direct product since $\;P\cap Q=\{1\}\;$ , and from all this it follows that
$$G=PQ=P\times Q$$
and $\;G\;$ is thus always abelian. Since there are two groups up to isomorphism of order $\;p^2\;$ for any prime $\;p\;$ , we get two unique (up to isomorphism) different groups of order $\;45\;$ (both abelian, again):
$$G_1=C_9\times C_5\;\;,\;\;\;G_2=C_3\times C_3\times C_5\cong C_3\times C_{15}$$
The above is a particular case of the general: if $\;p<q\;$ are two primes such that $\;p\,\nmid\,q-1\;$ , then there are two unique groups of prder $\;p^2\cdot q\;$ up to isomorphism
DonAntonio
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@lowiq We have that $;P\cap Q=1;$ because any element in the intersection must divide both $;p;$ and $;q;$ , and these two are different prime numbers... About more exercises: any book in group theory surely will contain some exercises of this, and in the internet there must be thousands of sites... – DonAntonio Jun 02 '16 at 16:18